reserve X for MetrSpace,
  x,y,z for Element of X,
  A for non empty set,
  G for Function of [:A,A:],REAL,
  f for Function,
  k,n,m,m1,m2 for Nat,
  q,r for Real;
reserve X for non empty MetrSpace,
  x,y for Element of X,
  V for Subset of X,
  S,S1,T for sequence of X,
  Nseq for increasing sequence of NAT;

theorem Th15:
  S is_convergent_in_metrspace_to x implies for r st 0 < r holds
  Ball(x,r) contains_almost_all_sequence S
proof
  assume
A1: S is_convergent_in_metrspace_to x;
  thus for r st 0 < r holds Ball(x,r) contains_almost_all_sequence S
  proof
    let r such that
A2: 0 < r;
    ex m st for n st m <= n holds S.n in Ball(x,r)
    proof
      consider m1 such that
A3:   for n st m1 <= n holds dist(S.n,x) < r by A1,A2;
      take k = m1;
      now
        let n;
        assume k <= n;
        then dist(x,S.n) < r by A3;
        hence S.n in Ball(x,r) by METRIC_1:11;
      end;
      hence thesis;
    end;
    hence thesis;
  end;
end;
